A305493 A binary encoding of the decimal representation of a number: for any number n >= 0, consider its decimal representation and replace each 9 with "111111111" and each other digit d with a "0" followed by d "1"s and interpret the result as a binary string.

0, 1, 3, 7, 15, 31, 63, 127, 255, 511, 2, 5, 11, 23, 47, 95, 191, 383, 767, 1023, 6, 13, 27, 55, 111, 223, 447, 895, 1791, 2047, 14, 29, 59, 119, 239, 479, 959, 1919, 3839, 4095, 30, 61, 123, 247, 495, 991, 1983, 3967, 7935, 8191, 62, 125, 251, 503, 1007, 2015

Offset

0,3

Comments

This sequence is a permutation of the nonnegative numbers.

The inverse sequence, say b, satisfies b(n) = A001202(n+1) for n = 0..1022, but b(1023) = 19 whereas A001202(1024) = 10.

The first known fixed points are: 0, 1, 65010; they all belong to A037308.

This encoding can be applied to any base b > 1 (when b = 2, we obtain the identity function) as well as to the factorial base and to the primorial base.

Links

Rémy Sigrist, Table of n, a(n) for n = 0..10000

Rémy Sigrist, Colored logarithmic scatterplot of the first 1000000 terms (where the color is function of the digital sum minus the number of nonleading digits different from 9)

Index entries for sequences related to decimal expansion of n

Index entries for sequences that are permutations of the natural numbers

Formula

A000120(a(n)) = A007953(n).

a(A051885(k)) = 2^k - 1 for any k >= 0.

a(A002275(k)) = A002450(k) for any k >= 0.

a(10 * n) = 2 * a(n).

Example

For n = 1972:
- the digit 1 is replaced by "01",
- the digit 9 is replaced by "111111111",
- the digit 7 is replaced by "01111111",
- the digit 2 is replaced by "011",
- hence we obtain the binary string "0111111111101111111011",
- and a(1972) = 2096123.

Mathematica

tb=Table[n->PadRight[{0}, n+1, 1], {n, 9}]/.PadRight[{0}, 10, 1]-> PadRight[ {}, 9, 1]; Table[FromDigits[IntegerDigits[n]/.tb//Flatten, 2], {n, 0, 60}] (* Harvey P. Dale, Jul 31 2018 *)

Prog

(PARI) a(n, base=10) = my (b=[], d=digits(n, base)); for (i=1, #d, if (d[i]!=base-1, b=concat(b, 0)); b=concat(b, vector(d[i], k, 1))); fromdigits(b, 2)
/* inverse */ b(n, base=10) = my (v=0, p=1); while (n, my (d = min(valuation(n+1, 2), base-1)); v += p * d; n \= 2^min(base-1, 1+d); p *= base); v

Crossrefs

Cf. A000120, A001202, A002275, A002450, A007953, A037308, A051885.

Sequence in context: A105754 A116690 A267258 * A213247 A043755 A105755

Adjacent sequences: A305490 A305491 A305492 * A305494 A305495 A305496

Keyword

nonn,base,easy

Author

Rémy Sigrist, Jun 02 2018

Status

approved